# P = Q MR = Q

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1 1. A firm operating in a monopolistically competitive market faces demand and marginal revenue curves as given below: The firm's total and marginal cost curves are: P = Q MR = Q TC = - 10Q Q MC = Q 2, where P is in dollars per unit, output rate Q is in units per time period, and total cost C is in dollars. Determine the price and output rate that will allow the firm to maximize profit or minimize losses. Compute a Lerner index. Calculate MR and equate it to MC. MC = MR Q 2 = Q 0.1Q Q - 20 = 0 The quadratic formula yields: Q 1 = Q 2 = Use Q 1 since negative quantities are not meaningful. At Q 1 = P = (13.17) = 8.68 Computation of monopoly power. The Lerner index is computed below: L P = MC P At Q = 13.17, P = 8.68, and MC = 7.34 L = ( )/8.68 = L = =

2 2. Bartels and Jaymes are two individuals who one day discover a stream that flows wine cooler instead of water. Bartels and Jaymes decide to bottle the wine cooler and sell it. The marginal cost of bottling wine cooler and the fixed cost to bottle wine cooler are both zero. The market demand for bottled wine cooler is given as: P = Q where Q is the total quantity of bottled wine cooler produced and P is the market price of bottled wine cooler. What is the economically efficient price of bottled wine cooler? What is the economically efficient quantity of bottled wine cooler produced? c. If Bartels and Jaymes were to collude with one another and produce the profitmaximizing monopoly quantity of bottled wine cooler, how much bottled wine cooler will they produce? d. Given the output level in (c), what price will Bartels and Jaymes charge for bottled wine cooler? e. At the output level in (c), what is the welfare loss? f. Suppose that Bartels and Jaymes act as Cournot duopolists, what are the reaction functions for Bartels and for Jaymes? g. In the long run, what level of output will Bartels produce if Bartels and Jaymes act as Cournot duopolists? h. In the long run, what will be the price of wine coolers be if Bartels and Jaymes act as Cournot duopolists? Suppose that after Bartels and Jaymes have arrived at their long run equilibrium as Cournot duopolists, another individual, Paul Mason, discovers the streams. Paul Mason, who will sell no wine cooler before its time, decides to bottle wine coolers. There are now three Cournot firms producing at once. In the long run, what level of output will Bartels produce? c. The economically efficient level of price is found where price equals marginal cost. The marginal cost is zero. Therefore, the efficient price is zero. At a price of zero, Q = 360. The profit maximizing level of output is found where MR = MC. The MR curve has the same price intercept as the demand curve and is twice as steep. Thus, a monopolist will produce half as much as the competitive level (this is only true because marginal cost is constant). The competitive level of output is 360. Therefore, the monopoly level of output is 180. Mathematically, the marginal revenue curve is: MR = Q Equating MR to MC yields: Q = 0 Q = 180

3 d. e. f. When Q = 180 we have P = (180) = 45. The welfare loss is the value of the output that would have been produced under the conditions of economic efficiency, but is not produced due to the monopoly. This is the area of the triangle from Q = 180 to Q = 360, under the demand curve. The base of the triangle is 180, the height of the triangle is 45, and therefore the welfare loss is (1/2)(180)(45) = 4,050. The Cournot equilibrium is found by using the reaction curves of the two firms to solve for levels of output. The reaction curve for firm 1 (Bartels) is found as follows: R 1 = PQ 1 = ( Q)Q 1 = 90Q (Q 1 + Q 2 )Q 1 = 90Q Q Q 2 Q 1 The firm's marginal revenue MR 1 is just the incremental revenue dr 1 resulting from an incremental change in output dq 1 : MR 1 = dr 1 /dq 1 = Q Q 2 Setting MR 1 equal to zero (the firm's marginal cost) and solving for Q 1 yields the reaction curve for Q 1 : Firm 1's Reaction Curve: Q 1 = (1/2)Q 2 Going through the same calculations for firm 2 yields: Firm 2's Reaction Curve: Q 2 = (1/2)Q 1 g. h. i. Solving the reaction curves simultaneously for Q 1 and Q 2 yields: Q 1 = Q 2 = 120. The total output is 240 (120 by each firm). Therefore: P = (240) = 30. The Cournot equilibrium for three firms is found by solving the reaction curves of the three firms simultaneously for levels of output. The reaction curve for firm 1 (Bartels) is found as follows: R 1 = PQ 1 = ( Q)Q 1 = 90Q (Q 1 + Q 2 + Q 3 )Q 1 = 90Q Q Q 2 Q Q 3 Q 1 The firm's marginal revenue MR 1 is just the incremental revenue dr 1 resulting from an incremental change in output dq 1 : MR 1 = dr 1 /dq 1 = Q Q Q 3 Setting MR 1 equal to zero (the firm's marginal cost) and solving for Q 1 yields the reaction curve for Q 1 : Firm 1's Reaction Curve: Q 1 = Q 2-0.5Q 3 Going through the same calculations for firm 2 yields:

4 Firm 2's Reaction Curve: Q 2 = Q 1-0.5Q 3 Going through the same calculations for firm 3 yields: Firm 3's Reaction Curve: Q 3 = Q 2-0.5Q 3 Solving the reaction curves simultaneously for Q 1, Q 2 and Q 3 yields: Q 1 = Q 2 = Q 3 = Lambert-Rogers Company is a manufacturer of petrochemical products. The firm's research efforts have resulted in the development of a new auto fuel injector cleaner that is considerably more effective than other products on the market. Another firm, G.H. Squires Company, independently developed a very similar product that is as effective as the Lambert- Rogers formul To avoid a lengthy court battle over conflicting patent claims, the two firms have decided to cross-license each other's patents and proceed with production. It is unlikely that other petrochemical companies will be able to duplicate the product, making the market a duopoly for the foreseeable future. Lambert-Rogers estimates the demand curve given below for the new cleaner. Marginal cost is estimated to be a constant \$2 per bottle. Q = 300,000-25,000P. where P = dollars per bottle and Q = monthly sales in bottles. Lambert-Rogers and G.H. Squires have very similar operating strategies. Consequently, the management of Lambert-Rogers believes that the Cournot model is appropriate for analyzing the market, provided that both firms enter at the same time. Calculate Lambert- Rogers' profit-maximizing output and price according to this model. Lambert-Rogers' productive capacity and technical expertise could allow them to enter the market several months before Squires. Choose an appropriate model and analyze the impact of Lambert Rogers being first into the market. Should Lambert-Rogers hurry to enter first? Denote Lambert-Rogers price and quantity as P L, Q L and Squires as P S, Q S. Demand function is given as: Q = 300,000-25,000P Solve for P: Q - 300,000 = -25,000P P = Q

5 Outcome under Cournot model: TR L = P L Q L TR L = ( Q)Q L Q = Q L + Q S TR L = [ (Q L + Q S )]Q L TR L = 12Q L Q 2 L Q L Q S MR L = Q L Q S Set MR L = MC Q L Q S = Q L Q S = -10 Q L = 125, Q S So, Q S = 125, Q L Substitute for Q S : Q L = 62, Q L Q L = 62, = 83,333 Q = Q L + Q S Q = 83, ,333 = 166,666 P = (166,666) P = = \$5.33 P = \$5.33 per bottle 166,666 bottles sold per month The Stackelberg model is appropriate when one firm enters first. Lambert-Rogers determines its output, which Squires then takes as given. Lambert's total revenue function is given as: TR L = 12Q L Q L Q L Q S

6 Squires reaction function Q S = 125, Q L can be substituted into TR L, since Squires will take Lambert's output as given. TR L = 12Q L Q L Q L (125, Q L ) TR L = 12Q L Q L 2-5Q L Q L 2 TR L = 7Q L Q L 2 MR L = Q L Set MR L = MC Q L = Q L = -5 Q L = 125,000 To find Q S substitute Q L into S reaction function Q S = 125, Q L Q S = 125, (125,000) Q S = 62,500 Q = Q L + Q S Q = 125, ,500 Q = 187,500 P = (187,500) P = = \$4.50 Lambert-Rogers gets a much larger share of the market by entering first. It should advance its schedule in order to enter first.

8 Q M = 140,000-32,000P Q F = 60, ,000P Denote dominant firm demand curve as Q D. Q D = Q M - Q F Q D = 140,000-32,000P - (60, ,000P) Q D = 80,000-40,000P Solve for P Q D - 80,000 = -40,000P P = Q D MR D = Q D Marginal cost for the dominant firm is \$0.75. Equate MR D to MC D Q D = Q D = Q D = 25,000 P = (25,000) P = = per gallon Fringe takes dominant firm price as given Q F = 60, ,000(1.375) Q F = 71,000 Q T = 25, ,000 = 96,000 A 40,000 increase in demand curve to: Q M = 180,000-32,000P Q F = 60, ,000P Q D = 180,000-72,000P - (60, P) Q D = 120,000-40,000P Solve for P Q D - 120,000 = -40,000P P D = Q D MR D = Q D setting MR D = MC D Q D = Q D = Q D = 45,000 P D = (45,000) P D = = \$1.875 Fringe again follows Q F = 60, ,000(1.875) Q F = 75,000

9 Q T = 45, ,000 = 120,000 We can see that when demand changed, the dominant firm raised price. The competitive fringe took the new price as given and adjusted output accordingly. 6. Consider two identical firms (no. 1 and no. 2) that face a linear market demand curve. Each firm has a marginal cost of zero and the two firms together face demand: P = Q, where Q = Q1 + Q2. Find the Cournot equilibrium Q and P for each firm. Find the equilibrium Q and P for each firm assuming that the firms collude and share the profit equally. c. Contrast the efficiencies of the markets in (a) and (b) above. Determine the reaction curve for no. 1. Equate MR 1 to MC 1. R 1 = P 1 Q 1 = (50-0.5Q)Q 1 = 50Q 1-0.5QQ 1 = 50Q 1-0.5(Q 1 + Q 2 )Q 1 = 50Q 1-0.5Q Q 1 Q 2 MR 1 = 50-1Q 1-0.5Q 2. Since MC 1 = 0, then 50-1Q 1-0.5Q 2 = 0 Q 1 = Q 2 The reaction curve for firm no. 2 is calculated in the same way as that for firm no. 1. Q 2 = Q 1 At the intersection of two reaction curves, we find the equilibrium Q 1 and Q 2. By substitution: Q 2 = (50-0.5Q 2 ) Q 2 = Q Q 2 = 25 Q 2 = 33.33

10 Now solve for Q 1 : Q 1 = Q 2 = (33.33) = The total quantity produced Q = Q 1 + Q 2 = The market equilibrium price is: P = 50-5Q = (66.67) = \$16.67/unit. Each firm is maximizing its own profit, given its competitor's production rate. The total revenue for the two firms is: R = PQ - (50-0.5Q)Q = 50Q - 0.5Q 2, and thus MR = 50 - Q Set MR equal to MC = 0 to find Q that maximizes profit Q = 0 Q = 50 If profit is shared equally, then Q 1 = Q 2 = 25. The collusive price is P = (50) = 25. c. When competing, each firm produces at a price of 16.67; and when colluding, each produces 25 at a price of 25. The market is more efficient when the firms compete, because in this situation selling price is more nearly equal to marginal cost. D D 7. The demand for on-line brokerage services is: Q = 6, P P = Q. If the online brokerage firms collude, the collusive marginal revenue function is: MR ( Q) = Q. MC1 ( q1 ) = 1.5q1 MC2 ( q2 ) = 2.0 q2 The brokerage firm specific marginal cost functions are:. Calculate the MC3 ( q3 ) = 2.5q3 MC4 ( q4 ) = 3.0q4 collusive output level and market price. If the brokerage firms behaved competitively and each firm set its own marginal cost equal to price, what would be the output level and market price?

11 2 q1 = MC 3 MC1 ( q1 ) = 1.5q 1 1 q2 = MC MC2 ( q2 ) = 2.0q2 2. MC3 ( q3 ) = 2.5q3 2 q3 = MC 4 ( 4 ) = 3.0 MC q q q4 = MC = = This implies the collusive supply sets S Q MC MC The collusive marginal cost function is: 10 MC =. 19 Q The collusive profit maximization level sets marginal cost equal to marginal revenue. This is: = = Q Q Q The resulting price is \$ If brokerage firms behave competitively, each firm sets marginal cost equal to price. Thus, 2 q1 = P 3 MC ( q ) = 1.5q 1 = MC ( q ) q MC3 ( q3 ) = 2.5q3 2 q3 = P 4 ( 4 ) = 3.0 MC q q q4 = P = = q2 P 2 2 = This implies competitive supply is S Q P P Setting supply equal to demand and solving for market price gives us the competitive solution. This is: 19 P = P P = The competitive market output level is

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